Vol. 9, No 1, 2011 pp. 65 - 70
DOI 10.2298/FUPCT1101065G
UDC 519.722
A NEW CONTINUOUS ENTROPY FUNCTION AND ITS INVERSE
Andrej Grebenc
Av. Brugmann 61, 1000 Brussels, Belgium
Abstract.A new continuous entropy function of the form h(x)=-k∫p'(x)*ln(p(x)) is constructed based on Lebesgue integral. Solution of the integral is of the form h(x)=-k*p(x)*ln(e-1*p(x)). Inverse solution of this function in the form of p(x)=h(x)*W-1(h(x)* e-1) has been obtained and this is a novelty. The fact that the integral of a logarithmic function is also a logarithmic function has been exploited and used in a more general ansatz for a deliberate function. The solution of the differential equation Λ(x)= g(x)'*ln(g(x)) for a deliberate function g(x) includes the Lambert W function. The solution is a parametric function g(x)=(s(x)+C)/W(e-1*(s(x)+C)). Parameter C has a minimal value and, consequently, an infimal function g(x) exists. This transform is of general use, and is particularly suitable for use in neural networks, the measurement of complex systems and economic modeling, since it can transform multivariate variables to exponential form.
Key words: sigmoid functions, logarithmic probability density and logarithmic cumulative probability functions, information entropy, solution of the inverse entropy problem fatigans
NOVA KONTINUALNA ENTROPIJSKA FUNKCIJA
I NJEN INVERZ
Nova funkcija neprekidne entropije oblika h(x)=-k∫p'(x)*ln(p(x)) definisana je pomoću Lebesgue-ovog integrala. Rešenje ovog integrala je oblika h(x)=-k*p(x)*ln(e-1*p(x)). Novina je da je pronađeno inverzno rešenje pomenute funkcije oblika p(x)=h(x)*W-1(h(x)* e-1. Činjenica da je integral logaritamske funkcije, takođe, logaritamska funkcija korišćena je u opštijem kontekstu vezanom za deliberate funkciju. Rešenje diferencijalne jednačine Λ(x)= g(x)'*ln(g(x)) za funkciju g(x) uključuje i Lambert-ovu W funkciju. Rešenje je parametarska funkcija g(x)=(s(x)+C)/W(e-1*(s(x)+C)). Kako parametar C ima minimalnu vrednost, postoji infimalna funkcija g(x).
Ovakav pristup ima široku primenu i može se koristiti kod neuronskih mreža, merenja kompleksnih sistema i u ekonomskom modelovanju, s obzirom na to da se njime multivarijantne promenljive mogu transformisati u eksponencijalni oblik.
Ključne reči: sigmoidna funkcija, logaritamska funkcija gustine i logaritamska funkcija zbirne verovatnoće, informaciona entropija, rešenje inverznog problema entropije
