Description Usage Arguments Details Value Author(s) References See Also Examples

Function `cxds.stheorem`

is applied to a pair of probability vectors or to a pair of vector of counts associated with two states of the same open thermodynamic system in order to estimate renormalized entropy shift as the system evolves.

1 | ```
cxds.stheorem(distribution0, distribution1)
``` |

`distribution0` |
vector of counts representing distribution function of state0 of the (open thermodynamic) system |

`distribution1` |
vector of counts representing distribution function of state1 of the (open thermodynamic) system |

The function implements the second part of the algorithm based on Klimontovich's S-theorem (the first part is implemented by `crit.stheorem`

). More detailes can be found in Klimontovich (1989) and Anischenko (1994).

Let the open thermodynamic system be characterized by probability density function *f(x)* and evolving from state0 to state1. Entropies corresponding to state0 and state1 are respectively:

*H_{0} = -\int {f_{0}(x)lnf_{0}(x)dx}, H_{1} = -\int {f_{1}(x)lnf_{1}(x)dx}*

This is known Shannon entropy formulation which is most comprehensive indicator of degree of order of the system at the certain state. Shannon entropy shift depends on both external (energy/entropy in-/out-flows) and internal (noise and self-organization processes) changes happened to the system. However, it doesn't make it possible to differentiate between two and estimate, for instance, the role of self-organization processes in the evolution. The quantifier proposed by Klimontovich (1987) is called renormalized entropy

*dS_{0\rightarrow 1} = -\int {f_{1}(x)lnf_{1}(x)dx} + \int {f^{*}(x)ln f^{*}(x)dx}*

where *f^{*}(x)* is a new probability density function being the result of Gibbs distribution function tranformation of *f_{0}(x)* for energy balancing (see the 'details' section for `crit.stheorem`

). The formula allows for the final entropy shift to be written in a form

*dH_{0\rightarrow 1}=H_{1}-H_{0} = dS_{0\rightarrow 1} + dI_{0\rightarrow 1}*

where *dI* is the entropy change caused by external reasons. If direct evolution from the state0 to state1 is thermodynamically forbidden (*r^{2}<1*) the medium state2 of the system evolution is to be found and the final formula for entropy shift will contain additional equation:

*dS_{0\rightarrow 1} = dS_{0\rightarrow 2}+dS_{2\rightarrow 1}*

representing the pass of the system through an indirect medium state.

It is recommended to apply the function alongside with Klimontovich's S-theorem convergence criterion function `crit.stheorem`

`dH_val ` |
Shannon entropy shift |

`dS_val ` |
renormalized entropy shift |

`dH_ext ` |
entropy inflow to the system from outside |

`dS_02 ` |
renormalized entropy shift from state0 to medium state2 |

`dS_21 ` |
renormalized entropy shift from medium state2 to state1 |

Vitaly Efremov <[email protected]>

Y.L.Klimontovich. S-theorem. Zeitschrift fur Physik B Condensed Matter. 1987, Volume 66, Issue 1, pp 125-127.

Yu.L.Klimontovich. Problems of open system statistical theory: criteria of relative state ordering at processes of self-organization. 1989. Usp. Fiz. Nauk. v.158(1) (in Russian)

V.S.Anishchenko, T.G.Anishchenko. On the criterion of the relative degree of order for self-oscillating regimes. Illustration of Klimontovich's S-theorem. Proc.SPIE, v.2098, pp.130-136, 1994.

K.Kopizki, P.C.Warnke, P.Saparin, j.Kurths, J.Timmer. Comment on 'Kullback-Leiber and renormalized entropies: Application to electroencephalograms of epilepsy patients'. PHYSICAL REVIEW E 66, 043902 (2002).

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 | ```
#quazi-gaussian probability vectors with equal means & different variances
f0 <- c(0.0,0.1,0.4,0.4,0.1,0.0)
f1 <- c(0.1,0.15,0.25,0.25,0.15,0.1)
cxds.stheorem(distribution0=f0, distribution1=f1)
#quazi-gaussian bin counts with shift between means
h0 <- c(2,2,17,6,1,1,1,0)
h1 <- c(2,3,5,7,7,4,1,0)
crit.stheorem(h0, h1)
cxds.stheorem(h0, h1)
#example of 2-step analysis with Klimontovich's S-theorem for 2
# arrays of outcomes {s0,s1}:
s0<-rep(c(1:11,2),256)
s1<-rep(c(2,3,3,4,5,5,5),55)
# step a. Create probability vectors
b<-d1nat(s0,s1,brks=12); b
# step b. Compare samples with Klimontovich's S-theorem
crit.stheorem(b$f0,b$f1)
cxds.stheorem(b$f0,b$f1)
#example of 3-step analysis with Klimontovich's S-theorem to study two gratings
# random vs regular
s0<-array(c(rep(0,640),rep(1,640)), c(320,320))
s1<-array(runif(5120,0,1), c(64,80))
# step a. Binarize (to make s1 comparable with s0 by its nature as a grating)
a<-utild2bin(s0, s1, method='med')
# step b. Create probability vectors as for angular space (anisotropy study)
# There is no doubt s0 is more regular
b<-d2spec(s0, s1, brks=36, method='ang90'); b
# step c. Compare gratings with Klimontovich's S-theorem. Renormalized entropy shift
# is negligible compared to Shannon's. Evolution from state0 to state1 is possible
# but clearly with external entropy (or energy) inflow
crit.stheorem(b$f0,b$f1)
cxds.stheorem(b$f0,b$f1)
#example of 2-step analysis with Klimontovich's S-theorem for
#two following char arrays:
s0<-c("a","b",rep("c",9),rep("d",2),"e","f","g",rep("h",2),"i","j"); s0
s1<-c(rep("a",16), rep("c",35), rep("i",13)); s1
# step a. Create probability vectors. It seems that s0 has lower level
# of orderliness (Shannon entropy is higher)
b<-d1char.d1nat(s0,s1); b
# step b. Compare samples with Klimontovich's S-theorem. Renormalized entropy indicates
# the opposite: s0 is more ordered and difference in Shannon entropy values was
# due to just "thermodynamic noise"
crit.stheorem(b$f0,b$f1)
cxds.stheorem(b$f0,b$f1)
#example of 3-step analysis with Klimontovich's S-theorem for 2 random
# arrays of outcomes {s0,s1}:
s0<-runif(128,0,1)^2
s1<-runif(64,0,1)^2.3
# step a. Convert samples to arrays of sequential 17-point means
a<-utild1group(s0, s1, radius=8, method='splitN')
# step b. Create probability vectors
b<-d1nat(a$group0,a$group1,brks=12,band=c(0,0.8)); b
# step c. Compare samples with Klimontovich's S-theorem
crit.stheorem(b$f0,b$f1)
cxds.stheorem(b$f0,b$f1)
``` |

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