arXiv subjects

Federico Zertuche

Publications and source records attributed to Federico Zertuche.

7 recordsLinked to original sources

arXiv2019

A Parametric Bootstrap for the Mean Measure of Divergence

For more than $50$ years the {\it Mean Measure of Divergence} (MMD) has been one of the most prominent tools used in anthropology for the study of non-metric traits. However, one of the problems, in anthropology including palaeoanthropology (more often there), is the lack of big enough samples or the existence of samples without sufficiently measured traits. Since 1969, with the advent of bootstrapping techniques, this issue has been tackled successfully in many different ways. Here, we present a parametric bootstrap technique based on the fact that the transformed $ θ$, obtained from the Anscombe transformation to stabilize the variance, nearly follows a normal distribution with zero mean and variance $ σ^2 = 1 / (N + 1/2) $, where $ N $ is the size of the measured trait. When the probabilistic distribution is known, parametric procedures offer more powerful results than non-parametric ones. We profit from knowing the probabilistic distribution of $ θ$ to develop a parametric bootstrapping method. We explain it carefully with mathematical support. We give examples, both with artificial data and with real ones. Our results show that this parametric bootstrap procedure is a powerful tool to study samples with scarcity of data.

stat.AP↗

arXiv2014

The Role of Boolean Irreducibility in \textit{NK}-Kauffman Networks

Boolean variables are such that they take only values on $ \mathbb{Z}_2 \cong \left\{0, 1 \right\} $. \textit{NK}-Kauffman networks are dynamical deterministic systems of $ N $ Boolean functions that depend only on $ K \leq N $ Boolean variables. They were proposed by Kauffman as a first step to understand cellular behaviour [Kauffman, S.A.; {\rm The Large Scale Structure and Dynamics of Gene Control Circuits: An Ensemble Approach}. {\it J. Theoret. Biol.} {\bf 44} (1974) 167.] with great success. Among the problems that still have been not well understood in Kauffman networks, is the mechanism that regulates the phase transition of the system from an ordered phase; where small changes of the initial state decay, to a chaotic, where they grow exponentially. We show, that this mechanism is regulated through \textsf{the irreducible decomposition} of Boolean functions proposed in [ Zertuche, F. {\rm On the robustness of NK-Kauffman networks against changes in their connections and Boolean functions}, {\it J. Math. Phys.} {\bf 50} (2009) 043513]. This is in contrast to previous knowledge that attributed it to \textsf{canalization}. We also review other statistical properties of Kauffman networks that have been shown that \textsf{Boolean irreducibility} explains.

nlin.AO↗

arXiv2014

Phase Transition in NK-Kauffman Networks and its Correction for Boolean Irreducibility

In a series of articles published in 1986 Derrida, and his colleagues studied two mean field treatments (the quenched and the annealed) for \textit{NK}-Kauffman Networks. Their main results lead to a phase transition curve $ K_c \, 2 \, p_c \left( 1 - p_c \right) = 1 $ ($ 0 < p_c < 1 $) for the critical average connectivity $ K_c $ in terms of the bias $ p_c $ of extracting a "$1$" for the output of the automata. Values of $ K $ bigger than $ K_c $ correspond to the so-called chaotic phase; while $ K < K_c $, to an ordered phase. In~[F. Zertuche, {\it On the robustness of NK-Kauffman networks against changes in their connections and Boolean functions}. J.~Math.~Phys. {\bf 50} (2009) 043513], a new classification for the Boolean functions, called {\it Boolean irreducibility} permitted the study of new phenomena of \textit{NK}-Kauffman Networks. In the present work we study, once again the mean field treatment for \textit{NK}-Kauffman Networks, correcting it for {\it Boolean irreducibility}. A shifted phase transition curve is found. In particular, for $ p_c = 1 / 2 $ the predicted value $ K_c = 2 $ by Derrida {\it et al.} changes to $ K_c = 2.62140224613 \dots $ We support our results with numerical simulations.

nlin.CG↗

arXiv2012

Z_2-Algebras in the Boolean Function Irreducible Decomposition

We develop further the consequences of the irreducible-Boolean classification established in Ref. [9]; which have the advantage of allowing strong statistical calculations in disordered Boolean function models, such as the \textit{NK}-Kauffman networks. We construct a ring-isomorphism $ mathfrak{R}_K {i_1, ..., i_λ} \cong \mathcal{P}^2 -[K] $ of the set of reducible $K$-Boolean functions that are reducible in the Boolean arguments with indexes ${i_1, ..., i_λ}$; and the double power set $\mathcal{P}^2 [K]$, of the first $K$ natural numbers. This allows us, among other things, to calculate the number $\varrho_K (λ, ω)$ of $K$-Boolean functions which are $λ$-irreducible with weight $ω$. $\varrho_K (λ, ω)$ is a fundamental quantity in the study of the stability of \textit{NK}-Kauffman networks against changes in their connections between their Boolean functions; as well as in the mean field study of their dynamics when Boolean irreducibility is taken into account.

math-ph↗

arXiv2009

On the Robustness of NK-Kauffman Networks Against Changes in their Connections and Boolean Functions

NK-Kauffman networks {\cal L}^N_K are a subset of the Boolean functions on N Boolean variables to themselves, Λ_N = {ξ: \IZ_2^N \to \IZ_2^N}. To each NK-Kauffman network it is possible to assign a unique Boolean function on N variables through the function Ψ: {\cal L}^N_K \to Λ_N. The probability {\cal P}_K that Ψ(f) = Ψ(f'), when f' is obtained through f by a change of one of its K-Boolean functions (b_K: \IZ_2^K \to \IZ_2), and/or connections; is calculated. The leading term of the asymptotic expansion of {\cal P}_K, for N \gg 1, turns out to depend on: the probability to extract the tautology and contradiction Boolean functions, and in the average value of the distribution of probability of the Boolean functions; the other terms decay as {\cal O} (1 / N). In order to accomplish this, a classification of the Boolean functions in terms of what I have called their irreducible degree of connectivity is established. The mathematical findings are discussed in the biological context where, Ψis used to model the genotype-phenotype map.

q-bio.QM↗

arXiv2007

The Number of Different Binary Functions Generated by NK-Kauffman Networks and the Emergence of Genetic Robustness

We determine the average number $ \vartheta (N, K) $, of \textit{NK}-Kauffman networks that give rise to the same binary function. We show that, for $ N \gg 1 $, there exists a connectivity critical value $ K_c $ such that $ \vartheta(N,K) \approx e^{ϕN} $ ($ ϕ> 0 $) for $ K < K_c $ and $\vartheta(N,K) \approx 1 $ for $ K > K_c $. We find that $ K_c $ is not a constant, but scales very slowly with $ N $, as $ K_c \approx \log_2 \log_2 (2N / \ln 2) $. The problem of genetic robustness emerges as a statistical property of the ensemble of \textit{NK}-Kauffman networks and impose tight constraints in the average number of epistatic interactions that the genotype-phenotype map can have.

nlin.AO↗

arXiv2003

The Asymptotic Number of Attractors in the Random Map Model

The random map model is a deterministic dynamical system in a finite phase space with n points. The map that establishes the dynamics of the system is constructed by randomly choosing, for every point, another one as being its image. We derive here explicit formulas for the statistical distribution of the number of attractors in the system. As in related results, the number of operations involved by our formulas increases exponentially with n; therefore, they are not directly applicable to study the behavior of systems where n is large. However, our formulas lend themselves to derive useful asymptotic expressions, as we show.

physics.bio-ph↗