Supplementary Material: Bandt-Pompe Symbolization Dynamics for Time Series with Tied Values: a Data-Driven Approach

Supplementary Material: Bandt-Pompe symbolization dynamics for time series with tied values: a data-driven approach.

Francisco Traversaro 1, Francisco O. Redelico 2,3, Marcelo Risk 2,4, Alejandro C. Frery 2 and Osvaldo A. Rosso 2,6,7

1 - Grupo de Investigación en sistemas de información, Universidad Nacional de Lanús, & CONICET, 29 de Septiembre 3901, B1826GLC Lanús, Buenos Aires, Argentina. 2 - Departamento de Informática en Salud, Hospital Italiano de Buenos Aires & CONICET, C1199ABB Ciudad Autónoma de Buenos Aires, Argentina. 3 - Universidad Nacional de Quilmes, Departamento de Ciencia y Tecnolog´ıa, Roque Sáenz Peña 352, B1876BXD Bernal, Buenos Aires, Argentina. 4 - Instituto Tecnólgico de Buenos Aires (ITBA) & CONICET, Av. Eduardo Madero 399, C1181ACH Ciudad Autónoma de Buenos Aires, Argentina. 5 - Laboratório de Computa¸cão Cient´ıfica e Análise Numérica, Universidade Federal de Alagoas, Av. Lourival Melo Mota, s/n, 57072-970 Maceió, AL, Brazil. 6 - Instituto de F´ısica, Universidade Federal de Alagoas Av. Lourival Melo Mota, s/n, 57072-970 Maceió, AL, Brazil. 7 - Complex Systems Group, Facultad de Ingenier´ıa y Ciencias Aplicadas, Universidad de los Andes, Las Condes, 12455 Santiago, Chile. E-mail: [email protected] ∗ 1 Considered chaotic systems and their parameters 1.1 Noninvertible maps The nonivertible maps considered, their parameters, initial condition and the corresponding Lyapunov exponent, are listed below (see Ref. Sprott (2003)). We also report the corresponding minimum embedding Dmin-value such that forbidden patterns cannot appear for D < Dmin (see Ref. Rosso et al. (2013)).

Logistic Map : • Xn = ρ Xn ( 1 Xn ) . (1) +1 − Parameter value: ρ = 4; initial condition: X0 =0.1; Lyapunov exponent: λ =ln2=0.693147181 ... Ref: May (1976). Associated minimum embedding, Dmin = 3.

Sine Map : • Xn+1 = A sin πXn . (2)

Parameter value: A = 1; initial condition: X0 =0.1; Lyapunov exponent: λ 0.689067. Ref: Strogatz (1994). ' Associated minimum embedding, Dmin = 3. 2

Tent Map : • Xn = A Min Xn, 1 Xn . (3) +1 { − } Parameter value: A = 2; initial condition: X0 = 1/√2; Lyapunov exponent: λ = ln A = 0.693147181 ... | | Ref: Devaney (1984). Associated minimum embedding, Dmin = 3.

Linear Congruential Generator : •

Xn+1 = A Xn + B Xn ( Mod C ) . (4)

Parameter values: A = 7141, B = 54773, C = 259200; initial condition: X0 = 0; Lyapunov exponent: λ = ln A =8.873608101 ... Ref: Knuth (1997)| | Associated minimum embedding, Dmin 7. ≥

Cubic Map : • 2 Xn = A Xn ( 1 X ) . (5) +1 − n Parameter value: A = 3; initial condition: X0 =0.1; Lyapunov exponent: λ 1.0986122883. Ref: Zeng et al. (1985). ' Associated minimum embedding, Dmin = 5.

Ricker’s Population Model : • −Xn Xn+1 = A Xn e . (6)

Parameter value: A = 20; initial condition: X0 =0.1; Lyapunov exponent: λ 0.384846. Ref: Ricker (1954). ' Associated minimum embedding, Dmin = 3.

Gauss Map : • Xn+1 = 1/Xn ( Mod 1 ) . (7)

Initial condition: X0 = 1; Lyapunov exponent: λ 2.373445. Ref: van Wyk and Steeb (1997). ' Associated minimum embedding, Dmin 7. ≥

Cusp Map : • Xn = 1 A Xn . (8) +1 − | | p Parameter value: A = 2; initial condition: X0 =0.5; Lyapunov exponent: λ =0.5. Ref: Beck and Schl¨ogl (1995). Associated minimum embedding, Dmin = 3.

Pinchers Map : • Xn = tanh S ( Xn C ) . (9) +1 | − | Parameter values: S = 2, C =0.5; initial condition: X0 = 0; Lyapunov exponent: λ 0.467944. Ref: Potapov and Ali (2000). ' 3

Associated minimum embedding, Dmin = 3.

Spence Map : • Xn = ln Xn . (10) +1 | | Initial condition: X0 =0.5; Lyapunov exponent: λ . Ref: Shaw (1981). →∞ Associated minimum embedding, Dmin = 3.

Sine–Circle Map : • K Xn = Xn +Ω sin2πXn ( Mod 1 ) . (11) +1 − 2π

Parameter values: Ω = 0.5, K = 2; initial condition: X0 =0.1; Lyapunov exponent: λ 0.353863. Ref: Arnold (1965). ' Associated minimum embedding, Dmin = 4.

1.2 Dissipative maps The dissipative maps considered, their parameters, initial condition and the corresponding Lyapunov exponents, are listed below (see Ref. Sprott (2003)). We also report the corresponding minimum embedding Dmin-value such that forbidden patterns cannot appear for D < Dmin (see Ref. Rosso et al. (2013)).

H´enon Map : • 2 Xn = 1 a X + b Yn +1 − n . (12) Yn+1 = Xn

Parameter values: a = 1.4, b = 0.3; initial conditions: X0 = 0, Y0 = 0.9; Lyapunov exponents: λ1 0.41922, λ2 1.62319. Ref:' H´enon (1976).' − Associated minimum embedding, X-component, Dmin = 3. Associated minimum embedding, Y -component, Dmin = 3.

Lozi Map : • Xn = 1 a Xn + b Yn +1 − | | . (13) Yn+1 = Xn Parameter values: a =1.7, b =0.5; initial conditions: X = 0.1, Y = 0.1; Lyapunov exponents: 0 − 0 λ1 0.47023, λ2 1.16338. Ref:' Lozi (1978).' − Associated minimum embedding, X-component, Dmin = 3. Associated minimum embedding, Y -component, Dmin = 3.

Delayed Logistic Map : • Xn = A Xn ( 1 Yn ) +1 − . (14) Yn+1 = Xn

Parameter values: A = 2.27; initial conditions: X0 = 0.001, Y0 = 0.001; Lyapunov exponents: λ 0.18312, λ 1.24199. 1 ' 2 ' − 4

Ref: Aronson et al. (1982). Associated minimum embedding, X-component, Dmin = 4. Associated minimum embedding, Y -component, Dmin = 4.

Tinkerbell Map : • 2 2 Xn = X Y + a Xn + b Yn +1 n − n . (15) Yn+1 = 2 Xn Yn + c Xn + d Yn Parameter values: a =0.9, b = 0.6, c = 2, d =0.5; initial conditions: X = 0, Y =0.5; Lyapunov − 0 0 exponents: λ1 0.18997, λ2 0.52091. Ref: Nusse and' Yorke (1994).' − Associated minimum embedding, X-component, Dmin = 4. Associated minimum embedding, Y -component, Dmin = 4.

Burgers’ Map : • 2 2 Xn = a X Y +1 n − n . (16) Yn+1 = b Yn + Xn Yn Parameter values: a =0.75, b =1.75; initial conditions: X = 0.1, Y =0.1; Lyapunov exponents: 0 − 0 λ1 0.12076, λ2 0.22136. Ref:' Whitehead and' − Macdonald (1984). Associated minimum embedding, X-component, Dmin = 4. Associated minimum embedding, Y -component, Dmin = 4.

Holmes Cubic Map : • Xn+1 = Yn 3 . (17) Yn+1 = b Xn + d Yn Y − − n Parameter values: b = 0.2, d = 2.77; initial conditions: X0 = 1.6, Y0 = 0; Lyapunov exponents: λ1 0.59458, λ2 2.20402. Ref:' Holmes (1979).' − Associated minimum embedding, X-component, Dmin = 5. Associated minimum embedding, Y -component, Dmin = 5.

Dissipative Standard Map : • X = X + Y ( Mod 2π ) n+1 n n+1 . (18) Yn+1 = b Yn + k sin(Xn) (Mod2π )

Parameter values: b = 0.1, k = 8.8; initial conditions: X0 = 0.1, Y0 = 0.1; Lyapunov exponents: λ1 1.46995, λ2 3.77254. Ref:' Schmidt and' Wang − (1985). Associated minimum embedding, X-component, Dmin 7. ≥ Associated minimum embedding, Y -component, Dmin 7. ≥

Ikeda Map : • Xn = γ + µ ( Xn cos φ Yn sin φ ) +1 − , (19) Yn+1 = µ ( Xn sin φ + Yn cos φ ) 5

where φ = β α / (1 + X2 + Y 2). − n n Parameter values: α = 6, β = 0.4, γ = 1, µ = 0.9; initial conditions: X0 = 0, Y0 = 0; Lyapunov exponents: λ1 0.50760, λ2 0.71832. Ref: Ikeda (1979).' ' − Associated minimum embedding, X-component, Dmin = 5. Associated minimum embedding, Y -component, Dmin = 5.

Sinai Map : • X = X + Y + δ cos2π Y ( Mod 1 ) n+1 n n n . (20) Yn+1 = Xn +2 Yn ( Mod 1 ) Parameter value: δ = 0.1; initial conditions: X = 0.5, Y = 0.5; Lyapunov exponents: λ 0 0 1 ' 0.95946, λ2 1.07714. Ref: Sinai (1972).' − Associated minimum embedding, X-component, Dmin = 5. Associated minimum embedding, Y -component, Dmin = 6.

Discrete Predator–Prey Map : •

Xn = Xn exp[ r (1 Xn/K ) α Yn ] ( Prey ) +1 − − . (21) Yn+1 = Xn [1 exp( α Yn) ] ( Predator ) − −

Parameter value: r = 3, K = 1, α = 5; initial conditions: X0 =0.5, Y0 =0.5; Lyapunov exponents: λ1 0.19664, λ2 0.03276. Ref:' Beddington 'et al. (1975). Associated minimum embedding, X-component, Dmin = 4. Associated minimum embedding, Y -component, Dmin = 4.

1.3 Conservative maps The conservative maps considered, their parameters, initial condition and the corresponding Lyapunov exponents, are listed below (see Ref. Sprott (2003)). We also report the corresponding minimum embedding Dmin-value such that forbidden patterns cannot appear for D < Dmin (see Ref. Rosso et al. (2013)).

Chirikov Standard Map : • X = X + Y ( Mod 2π ) n+1 n n+1 . (22) Yn+1 = Yn + k sin Xn ( Mod 2π )

Parameter value: k = 1; initial conditions: X0 = 0, Y0 = 6; Lyapunov exponents: λ1,2 0.10497. Ref: Chirikov (1979). ' ± Associated minimum embedding, X-component, Dmin = 4. Associated minimum embedding, Y -component, Dmin = 4.

H´enon Area–Preserving Quadratic Map : • 2 Xn+1 = Xn cos α ( Yn Xn ) sin α − − 2 . (23) Yn+1 = Xn sin α + ( Yn X ) cos α − n 6

Parameter value: cos α = 0.24; Initial conditions: X0 = 0.6, Y0 = 0.13; Lyapunov exponents: λ1,2 0.00643. Ref:' H´enon ± (1969). Associated minimum embedding, X-component, Dmin = 4. Associated minimum embedding, Y -component, Dmin = 4.

Arnold’s Cat Map : • X = X + Y ( Mod 1 ) n+1 n n . (24) Yn+1 = Xn + k Yn ( Mod 1 )

Parameter value: k = 2; initial conditions: X0 = 0, Y0 = 1/√2; Lyapunov exponents: λ1,2 = 1 √ ln[ 2 (3 + 5)] = 0.96242365 ... Ref:± Arnold and Avez± (1968). Associated minimum embedding, X-component, Dmin = 6. Associated minimum embedding, Y -component, Dmin = 6.

Gingerbreadman Map : • Xn = 1 + Xn Yn +1 | | − . (25) Yn+1 = Xn

Initial conditions: X0 =0.5, Y0 =3.7; Lyapunov exponents: λ1,2 0.07339. Ref: Devaney (1984). ' ± Associated minimum embedding, X-component, Dmin = 4. Associated minimum embedding, Y -component, Dmin = 4.

Chaotic Web Map : •

Xn = Xn cos α ( Yn + k sinXn ) sin α +1 − . (26) Yn+1 = Xn sin α + ( Yn + k sinXn ) cos α

Parameter value: α = π/2, k = 1; initial conditions: X0 = 0, Y0 = 3; Lyapunov exponents: λ1,2 0.04847. Ref:' Chernikov ± et al. (1988). Associated minimum embedding, X-component, Dmin = 4. Associated minimum embedding, Y -component, Dmin = 4.

Lorenz–3D Chaotic Map : • Xn = Xn Yn Zn +1 −  Yn+1 = Xn . (27)  Zn+1 = Yn

Initial conditions: X0 =0.5, Y0 =0.5, Z0 = 1; Lyapunov exponents: λ1,2,3 0.07456, 0, 0.07456. Ref: Lorenz (1993). − ' − Associated minimum embedding, X-component, Dmin = 5. Associated minimum embedding, Y -component, Dmin = 5. Associated minimum embedding, Z-component, Dmin = 5. 7

2 Quantitative assessment of techniques for low-precision data

∗ ∗ T T Let xt t=1 and xt t=1 be the original and coarse time series, respectively, for which the permutationX ≡ { patterns} PDFsX ≡{ (embedding} dimension D and time lag τ) are Π∗ and Π. Consider Υ an operator over a permutation pattern PDF, e.g., the Shannon Entropy. We consider four different ways of perform the mapping to a symbolic alphabet for , which lead to Π(i), 1 i 4. The different imputation ways are: 1) Complete; 2) Random; 3) TimeX Ordered and 4) Data Driven.≤ ≤ Our hypothesis is that the operator applied to the best symbolic imputation technique will lead to the value closest to the original value. We thus want to find the technique i for which

Υ(Π(i)) Υ(Π∗) δ(i)Υ = | − | , (28) Υ(Π∗) is minimum. Notice that Eq. (28) is a measure of the deviation of from ∗, i.e., a measure of the error in which symbolic imputation i incurs. X X In order to gain a deeper understanding of the underlying mechanics, in the following we analyze the behavior of Υ(Π(i)) for two different operators Υ: the normalized Permutation Shannon Entropy (H) and the number of forbidden/missing patterns (NFMP), considering embedding dimension D = 3, 4, 5, 6 and time lag τ = 1. We evaluated times series ∗ of T = 105 iteration-data (after discarding 104 iterations for stability) from the chaotic maps presentedX in Sec. 1. They were produced with IEEE 754 double precision floating point numbers. The corresponding values for Υ(Π∗) were taken from Ref. Rosso et al. (2013). The ∗ coarse version (low precision) time series is obtained truncating the original values xt to two decimal digits, and then corresponding Υ(Π(i)) wasX computed. The obtained results for δH and δNFMP corresponding to the chaotic maps grouped by their Dmin (see Ref. Rosso et al. (2013)) are shown in the following figures: Figs. 1 chaotic maps with Dmin = 3; Figs. 2 chaotic maps with Dmin = 4; Figs. 3 chaotic maps with Dmin = 5; Figs. 4 chaotic maps with Dmin = 6; and Figs. 5 chaotic maps with Dmin 7. ≥ 8

Figure 1. Performance for the Logistic Map, Dmin = 3. 9

Figure 1. Performance for the Sine Map, Dmin = 3. 10

Figure 1. Performance for the Tent Map, Dmin = 3. 11

Figure 1. Performance for the Ricker’s Population Model, Dmin = 3. 12

Figure 1. Performance for the Cusp Map, Dmin = 3. 13

Figure 1. Performance for the Pincher’s Map, Dmin = 3. 14

Figure 1. Performance for the Spence Map, Dmin = 3. 15

Figure 1. Performance for the H´enon Map, Dmin = 3. 16

Figure 1. Performance for the Lozi Map, Dmin = 3. 17

Figure 2. Performance for the Sine–Circle Map, Dmin = 4. 18

Figure 2. Performance for the Delayed Logistic Map, Dmin = 4. 19

Figure 2. Performance for the Tinkerbell Map, Dmin = 4. 20

Figure 2. Performance for the Burgers’ Map, Dmin = 4. 21

Figure 2. Performance for the Discrete Predator–Prey Map, Dmin = 4. 22

Figure 2. Performance for the Chirikov Standard Map, Dmin = 4. 23

Figure 2. Performance for the H´enon Area–Preserving Quadratic Map, Dmin = 4. 24

Figure 2. Performance for the Gingerbreadman Map, Dmin = 4. 25

Figure 2. Performance for the Chaos Web Map, Dmin = 4. 26

Figure 3. Performance for the Cubic Map, Dmin = 5. 27

Figure 3. Performance for the Holmes Cubic Map, Dmin = 5. 28

Figure 3. Performance for the Ikeda Map, Dmin = 5. 29

Figure 3. Performance for the Lorenz 3D Chaotic Map, Dmin = 5. 30

Figure 3. Performance for the Lorenz 3D Chaotic Map, Dmin = 5. 31

Figure 3. Performance for the Sinai Map (X), Dmin = 5. 32

Figure 4. Performance for the Sinai Ma (Y), Dmin = 6. 33

Figure 4. Performance for the Arnold’s Cat Map, Dmin = 6. 34

Figure 5. Performance for the Linear Congruential Generator, Dmin ≥ 7. 35

Figure 5. Performance for the Gauss Map, Dmin ≥ 7. 36

Figure 5. Performance for the Dissipative Standard Map, Dmin ≥ 7. 37

References V. I. Arnold. Small denominators, I: mappings of the circumference into itself, American Mathemat- ical Society Translation Series 46 (1965) 213 – 284. V. I. Arnold, A. Avez. Ergodic problems of classical mechanics, Benjamin, New York, 1968. D. G. Aronson, M. A. Chory, G. R. Hall, R. P. McGehee. Bifurcations from an invariant circle for two- parameter families of maps of the plane: a computer-assisted study, Communications in Mathematical Physics 83 (1982) 304 – 354. C. Beck, F. Schlögl. Thermodynamics of chaotics systems, Cambridge University Press, New York, 1995. J. R. Beddington, C. A. Free, J. H. Lawton. Dynamic complexity in predador-prey models framed in difference equations, Nature 255 (1975) 58 – 60. A. A. Chernikov, R. Z. Sagdeev, G. M. Zaslavsky. Chaos: how regular can it be?, Physics Today 41 (1988) 27 – 35. B. V. Chirikov. A universal instability of many-dimensional oscillator systems, Physics Reports 52 (1979) 273 – 379. R. L. Devaney. An introduction to chaotic dynamical systems (2nd Edition), Addison-Wesley, Red- wood City, CA, 1989. M. Hénon. Numerical study of quadratic area-preserving mappings, Quarterly of Applied Mathematics 27 (1969) 291 – 312. M. Hénon. A two-dimensional mapping with a strange attractor, Communication on Mathematical Physics 50 (1976) 69 – 77. P. J. Holmes. A nonlinear oscillator with a strange attractor, Philosophical Transactions of the Royal Society of London Series A 292 (1979) 419 – 48. E. Ikeda. Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system, Optics Communications 30 (1979) 257 – 61. D. E. Knuth. Sorting and searching , Vol. 3 of The art of computer programming (3rd Edition), Addison-Wesley-Longman, Reading, MA, 1997. R. Lozi. Un attracteur étrange? Du type attracteur de Hénon, Journal de Physique 39 (1978) 9 – 10. E. N. Lorenz, The essence of chaos, University of Washington Press, Seattle, WA, 1993. R. May. Simple mathematical models with very complicated dynamics, Nature 261 (1976) 45 – 67. H. E. Nusse, J. A. Yorke. Dynamics: numerical explorations, Springer, New York, 1994. A. Potapov, M. K. Ali. Robust chaos in neural networks, Phys. Lett. A 277, 310 – 322 (2000). W. Ricker. Stock and recruitment, Journal of the Fisheries Research Board of Canada 11 (1954) 559 – 663. O. A. Rosso, F. Olivares, L. Zunino, L. De Micco, A. L. L. Aquino, A. Plastino, H. A. Larrondo Char- acterization of chaotic maps using the permutation Bandt-Pompe probability distribution, The European Physical Journal B (2013) 86: 116 G. Schmidt, B. H. Wang. Dissipative standard map, Physical Review A 32 (1985) 2994 – 2999. R. Shaw. Strange attractors, chaotic behaviour, and information flow, Zeitschrift für Naturforschung A 36 (1981) 80 – 112. Ya. G. Sinai. Gibbs measures in ergodic theory, Russian Mathematical Surveys 27 (1972) 21 – 69. J. C. Sprott. Chaos and time-series analysis, Oxford University Press, New York, 2003. S. H. Strogatz, Nonlinear dymanics and chaos with applications to physics, biology, chemistry, and engineering (Addison-Wesley-Longman, Reading, MA, 1994). M. A. van Wyk, W. Steeb. Chaos in electronics, Kluwer, Dordrecht, 1997. R. R. Whitehead, N. Macdonald. A chaotic mapping that displays its own homoclinic structure, Physica D 13 (1984) 401 – 407. W. Zeng, M. Ding, J. Li. Symbolic description of periodic windows in the antisymmetric cubic map, Chinese Physics Letter 2 (1985) 293 – 296.