Search Results (25)

In this paper, we investigate refined level sets associated with branching random walks on the boundary of a supercritical Galton–Watson tree. More precisely, for a prescribed deterministic sequence $\mathbf{s}:=(r_n,{\tilde{r}}_n)$, we introduce refined deviation sets, denoted by $E_{\mathbf{s}}(\alpha ,\beta )$, consisting of boundary points for which the deviations $S_n\mathsf{X}\left(t\right)-\alpha S_n\tilde{\mathsf{X}}\left(t\right)$ and $S_n\mathsf{Y}\left(t\right)-\beta S_n\tilde{\mathsf{Y}}\left(t\right)$ are asymptotically equivalent to $r_n$ and ${\tilde{r}}_n$, respectively, as $n\to \infty$. This setting extends the classical law-of-large-numbers level sets by allowing for controlled deviations around the typical linear behavior of the branching random walks. By constructing suitable inhomogeneous Mandelbrot measures, we establish sufficient conditions under which the sets $E_{\mathbf{s}}(\alpha ,\beta )$ have maximal Hausdorff and packing dimensions. Full article

East Coast Fever (ECF) causes approximately one million livestock deaths annually in sub-Saharan Africa, posing a significant threat to livestock. The wildlife–livestock interface complicates disease management, as wildlife serve as reservoirs. This study developed a Continuous Time Markov Chain (CTMC) model incorporating the wildlife–livestock interface to analyze ECF dynamics. Using the Galton–Watson approximation, we assessed the probability of disease extinction following the introduction of infected hosts or vectors. The probability of disease extinction calculated from the branching process is shown to be in good agreement with the probability approximated from numerical simulations. The disease dynamics of the deterministic model and the CTMC model are compared to ascertain the effect of demographic stochasticity on ECF dynamics. Differences in model predictions and asymptotic dynamics between stochastic and deterministic models were evident. The deterministic and stochastic formulations should therefore be viewed as complementary modeling frameworks, with the deterministic model characterizing average epidemic dynamics and the CTMC model capturing the probabilistic variability and extinction behavior inherent in real transmission processes. These differences are crucial for intervention strategies earmarked to prevent outbreaks. Our analysis revealed a high probability of ECF extinction if the disease emerges from recovered carrier cattle. Finite time to ECF disease extinction is estimated using 10,000 sample paths, and it is shown that the epidemic duration is shortest if the disease is introduced by infectious cattle. The epidemic duration is longest when the disease is introduced by infectious ticks. Additionally, we observed that host interactions at the wildlife–livestock interface play a critical role in shaping ECF transmission and informing control strategies. Full article

Today, it is common practice to distinguish something as intelligent or unintelligent based on its origin or behavior. One of the biggest discoveries of evolutionary biology is rapid evolution, which permeates every layer of the natural world. This is where natural glimpses of microevolutionary forms can be observed, revealing living organisms’ adaptive capacities converging towards intelligent behavior. In comparison, according to a Kantian postulate, encompassing ethical and anthropological conditions, nature acts for man until he is capable of acting with free intelligence; that is, until reason is fully realized to guide men towards performing morally good actions. This deliberation concerns humans acting with commendable conduct in a unified concept of will through reason to grasp not simply intelligence but a logical faculty that shapes our sense of duty. In Kant’s view of nature, this study posits in non-human animals’ signs of free intelligence in accidental relations with external agents, reaching an admirable display of ingenious abilities, as displayed in Kanzi and the South African beetle. Although it is difficult at times to distinguish purely reflex actions, humans’ reasoning strategies are not capable of reaching Kant’s practical maxims as a tool for achieving the greatest well-being necessary for all mankind. Full article

In both experimental and theoretical analyses of data, we often look to select a set of simple components that can be combined to create an appropriate model for the data. A convenient way to do this is to use quantile functions that can be added or transformed to obtain new distributions. In this work, we connect quantile statistics and mean absolute deviations (MADs) by deriving a general class of MAD-based shape metrics expressed as integrals of the quantile function, with a direct geometric interpretation. Our approach is applicable to distributions with finite mean that include many of the commonly used distributions, including those without a variance, such as the Pareto. When simple midpoint quadrature is used, the proposed metrics recover widely used quantile-shape metrics, including the interquartile range, Galton skewness, and Moore’s octile kurtosis as special cases. We further propose a C-Trapezoid quadrature approximation that combines cubic polynomial endpoint extrapolation with trapezoidal integration, achieving approximation errors that are significantly lower than those of the midpoint approximation for many common distributions. The proposed approximation provides simple-to-compute formulas for shape analysis and yields closed-form, non-iterative parameter-estimation formulas. These formulas are easy to compute and interpret, and they are applicable to a wide class of distributions, including those without an explicit cumulative distribution function or some with heavy tails. Unlike maximum likelihood estimation, the proposed method is more robust and has simple geometric interpretation. We illustrate the methodology with two detailed case studies. The proposed approach gives a simple way to quickly assess distributional shape without any specialized tools. Full article

Background: The complex interplay between treatment interventions and asymptomatic carriers and its effect on the epidemic duration of an infectious disease is not fully understood. Methods: Here, we used Galton-Watson branching process and generating function technique to estimate the density functions of minor outbreak duration. Simulations were used to calculate the central tendency of outbreak duration and address how changing levels of treatment failure affect this estimated duration. Results:Streptococcus pyogenes infection was used as a case study. Given the existence of the threshold, the change in mean duration as the probability of treatment failure increases is shown to be similar to the pattern driven by the basic reproduction number. In a supercritical regime, the mean duration tends to decrease as the probability of treatment failure increases. The distribution changes in tail behavior, from heavy- to light-tailed, if a large fraction of long extinction times develops to a major outbreak. Conclusions: Treatment failure elevates the probability of secondary transmissions by prolonging the overall infectious period, resulting in an extended the outbreak duration. The threshold of treatment failure identifies the maximum tolerable error for medical intervention. An unusually long period implies a critical early warning signal of a potential major outbreak that was successfully contained. Full article

One of the first serious attempts to study the indices that assess the association between the variables of a 2 × 2 contingency table was undertaken in the mid-1880s. Central to this study is the 1884 tornado observation/prediction data collected by Seargent John Park Finley (1854–1943), while working for the US Army Signal Service, and the controversial index he proposed to evaluate the success of his tornado predictions, which he denoted i. Subsequent improvements to Finley’s index were proposed, all of which pre-date the development of association measures made by pioneers such as Sir Francis Galton and Karl Pearson. This paper discusses Finley’s data, his index i, and the improvements made to this index. We also give historical context to Finley and his successors and their place in the early development of contingency table analysis. Full article

Classical branching-process theory, developed by Galton and Watson in the nineteenth century and later refined by Fisher and Haldane, provides the formal framework for quantifying the fate of new mutants, new viral and bacterial pathogens, new colonization of invasive species, etc. It is a powerful tool to quantify and predict the effect of differential reproductive success on the speciation potential of evolutionary lineages. Here, I revisit the conceptual framework of the branching process, detail its mathematical development over time, tie up a few historical loose strings, illustrate the calculation of the exact extinction probability for the Poisson-distributed reproductive success with the Lambert function (which is often missing in the ecological and evolutionary literature), and highlight the potential applications of the branching process in modern ecology and evolutionary biology, especially in deriving the extinction probability and extinction time. I also highlight a few misconceptions about human demography in the US that can be readily dismissed by applying probability tools such as branching processes. Full article

Let $\{Z_n:n\ge 0\}$ be a Galton–Watson system with a circular mechanism $\mathit{a}\ast \mathit{b}$, where $\mathit{a}={\left\{a_j\right\}}_{j=0}^{\infty }$ and $\mathit{b}={\left\{b_j\right\}}_{j=0}^{\infty }$ are probability distributions on ${\mathbf{Z}}_{+}:=\{0,1,2,\cdots \}$. Let $m_a:={\displaystyle \sum _{j=0}^{\infty }}j{a}_j$, $m_b:={\displaystyle \sum _{j=0}^{\infty }}j{b}_j$. The extinction property of such branching systems is first studied. Then, it is proved that there exists ${\Gamma }_n$ such that $W_n={\Gamma }_n^{-1}{Z}_n$ is an integrable martingale and hence converges to some random variable W. Moreover, for the case that $a_0=b_0=0, a_1,b_1\in (0,1)$, the convergence rates to 0 of $P\left(\left|{\displaystyle \frac{Z_{n+1}}{Z_n}}-{\omega }_n\right|>\epsilon \mid Z_0=1\right),P\left(\right|W_n-W|>\epsilon \mid Z_0=1)$ and $P\left(\left|{\displaystyle \frac{Z_{n+1}}{Z_n}}-{\omega }_n\right|>\epsilon \mid W\ge \delta ,Z_0=1\right)$ as $n\to \infty$ are presented for $\epsilon ,\delta >0$ under various moment conditions on $\left\{a_j\right\}$ and $\left\{b_j\right\}$, where ${\omega }_n=m_a$ or $m_b$ for n being even or odd, respectively. It is further shown that the first rate is geometric while the last two rates are supergeometric under a finite moment generating function hypothesis. Full article

In the present work, we consider three branching random walk $S_n\mathsf{Z}\left(\mathsf{t}\right),$$\mathsf{Z}\in \{\mathsf{X},\mathsf{Y},\Phi \}$ on a supercritical random Galton–Watson tree $\partial \mathsf{T}$. We compute the Hausdorff and packing dimensions of the level set ${\mathrm{E}}_{\chi }(\alpha ,\beta )=\left\{\mathsf{t}\in \partial \mathsf{T}:{\lim }_{n\to \infty }{\displaystyle \frac{S_n\mathsf{X}\left(\mathsf{t}\right)}{S_n\mathsf{Y}\left(\mathsf{t}\right)}}=\alpha \mathrm{and}{\lim }_{n\to \infty }{\displaystyle \frac{S_n\mathsf{Y}\left(\mathsf{t}\right)}{n}}=\beta \right\},$ where $\partial \mathsf{T}$ is endowed with random metric using $S_n\Phi \left(\mathsf{t}\right)$. This is achieved by constructing a suitable Mandelbrot measure supported on $\mathrm{E}(\alpha ,\beta )$. In the case where $\Phi =1$, we develop a formalism that parallels Olsen’s framework (for measures) and Peyrière’s framework (for the vectorial case) within our setting. Full article

Lifetime reproductive output (LRO), also called lifetime reproductive success (LRS) is often described by its mean (total fertility rate or net reproductive rate), but it is in fact highly variable among individuals and often positively skewed. Several approaches exist to calculating the variance and skewness of LRO. These studies have noted that a major factor contributing to skewness is the fraction of the population that dies before reaching a reproductive age or stage. The existence of that fraction means that LRO has a zero-inflated distribution. This paper shows how to calculate that fraction and to fit a zero-inflated Poisson or zero-inflated negative binomial distribution to the LRO. We present a series of applications to populations before and after demographic transitions, to populations with particularly high probabilities of death before reproduction, and a couple of large mammal populations for good measure. The zero-inflated distribution also provides extinction probabilities from a Galton-Watson branching process. We compare the zero-inflated analysis with a recently developed analysis using convolution methods that provides exact distributions of LRO. The agreement is strikingly good. Full article

Diffusion is a commonly used technique for spreading information from point to point on a graph. The rationale behind diffusion is not clear. The multi-type Galton–Watson forest is a random model of population growth without space or any other resource constraints. In this paper, we use the degenerated multi-type Galton–Watson forest (MGWF) to interpret the diffusion process, corresponding vertices to types and establishing an equivalence relationship between them. With the two-phase setting of the MGWF, one can interpret the diffusion process and the Google PageRank system explicitly. It also improves the convergence behavior of the iterative diffusion process and Google PageRank system. We validate the proposal by experiment while providing new research directions. Full article

The nonparametric bootstrap has been used in cluster analysis for various purposes. One of those purposes is to account for sampling variability. This can be achieved by obtaining a bootstrap approximation of the sampling distribution function of the estimator of interest and then clustering those distribution functions. Although the consistency of the nonparametric bootstrap in estimating transformations of the sample mean has been known for decades, little is known about how it carries over to clustering. Here, we investigated this problem with a simulation study. We considered single-linkage agglomerative hierarchical clustering and a three-type branching process for parametrized transformations of random vectors of relative frequencies of possible types of the index case of each process. In total, there were nine factors and 216 simulation scenarios in a fully-factorial design. The ability of the bootstrap-based clustering to recover the ground truth clusterings was quantified by the adjusted transfer distance between partitions. The results showed that in the best 18 scenarios, the average value of the distance was less than 20 percent of the maximum possible distance value. We noticed that the results most notably depended on the number of retained clusters, the distribution for sampling the prevalence of types, and the sample size appearing in the denominators of relative frequency types. The comparison of the bootstrap-based clustering results with so-called uninformed random partitioning results showed that in the vast majority of scenarios considered, the bootstrap-based approach led, on average, to remarkably lower classification errors than the random partitioning. Full article

Consider a supercritical Galton–Watson process with immigration $({\mathrm{X}}_n;n\ge 0)$. The Lotka–Nagaev estimator $\frac{{\mathrm{X}}_{n+1}}{{\mathrm{X}}_n}$ is a common estimator for the offspring mean. In this work, we used the Martingale method to establish several types of Cramér moderate deviation results for the Lotka–Nagaev estimator. To satisfy our needs, we employed the well-known Cramér approach for our proofs, which establishes the moderate deviation of the sum of the independent variables. Simultaneously, we provided a concrete example of its applicability in constructing confidence intervals. Full article

Let $\partial \mathsf{T}$ be a super-critical Galton–Watson tree. Recently, the first author computed almost surely and simultaneously the Hausdorff dimensions of the sets of infinite branches of the boundary of $\partial \mathsf{T}$ along which the sequence $S_n\mathsf{X}\left(t\right)/S_n\tilde{\mathsf{X}}\left(t\right)$ has a given set of limit points, where $S_n\mathsf{X}\left(t\right)$ and $S_n\tilde{\mathsf{X}}\left(t\right)$ are two branching random walks defined on $\partial \mathsf{T}$. In this study, we are interested in the study of the speed of convergence of this sequence. More precisely, for a given sequence $s=\left(s_n\right)$, we consider $E_{\alpha ,s}=\left\{t\in \partial \mathsf{T}:S_n\mathsf{X}\left(t\right)-\alpha S_n\tilde{\mathsf{X}}\left(t\right)\sim s_n\mathrm{as}n\to +\infty \right\}.$ We will give a sufficient condition on $\left(s_n\right)$ so that $E_{\alpha ,s}$ has a maximal Hausdorff and packing dimension. Full article

We consider a discrete-time single server queueing system, where arrivals stem from a multi-type Galton–Watson branching process with migration. This branching-type arrival process exhibits intricate correlation, and the performance of the corresponding queueing process can be assessed analytically. We find closed-form expressions for various moments of both the queue content and packet delay. Close inspection of the arrival process at hand, however, reveals that sample paths consist of large independent bursts of arrivals followed by geometrically distributed periods without arrivals. Allowing for non-geometric periods without arrivals, and correlated bursts, we apply $\pi$-thinning on the arrival process. As no closed-form expressions can be obtained for the performance of the corresponding queueing system, we focus on approximations of the main performance measures in the light and heavy traffic regimes. Full article