1 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 Investigaci´on en sistemas de informaci´on, Universidad Nacional de Lan´us, & CONICET, 29 de Septiembre 3901, B1826GLC Lan´us, Buenos Aires, Argentina. 2 - Departamento de Inform´atica en Salud, Hospital Italiano de Buenos Aires & CONICET, C1199ABB Ciudad Aut´onoma de Buenos Aires, Argentina. 3 - Universidad Nacional de Quilmes, Departamento de Ciencia y Tecnolog´ıa, Roque S´aenz Pe˜na 352, B1876BXD Bernal, Buenos Aires, Argentina. 4 - Instituto Tecn´olgico de Buenos Aires (ITBA) & CONICET, Av. Eduardo Madero 399, C1181ACH Ciudad Aut´onoma de Buenos Aires, Argentina. 5 - Laborat´orio de Computa¸c˜ao Cient´ıfica e An´alise Num´erica, Universidade Federal de Alagoas, Av. Lourival Melo Mota, s/n, 57072-970 Macei´o, AL, Brazil. 6 - Instituto de F´ısica, Universidade Federal de Alagoas Av. Lourival Melo Mota, s/n, 57072-970 Macei´o, 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 em- bedding 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 expo- nent: 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 em- bedding 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 em- bedding 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.