cxds.stheorem: Renormalized Entropy Shift Estimation

Description

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.

Usage

cxds.stheorem(distribution0, distribution1)

Arguments

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

Value

dH_valShannon entropy shift
dS_valrenormalized entropy shift
dH_extentropy inflow to the system from outside
dS_02renormalized entropy shift from state0 to medium state2
dS_21renormalized entropy shift from medium state2 to state1

Details

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="" system="" evolution="" is="" to="" be="" found="" and="" final="" formula="" for="" entropy="" shift="" will="" contain="" additional="" equation:="" $$ds_{0\rightarrow="" 1}="dS_{0\rightarrow" 2}+ds_{2\rightarrow="" 1}$$="" representing="" pass="" through="" an="" indirect="" state.<="" p="">

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

References

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).

Examples

Run this code

#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)

Run the code above in your browser using DataLab