scholarly journals On the number of vertices with a given degree in a Galton-Watson tree

2005 ◽  
Vol 37 (01) ◽  
pp. 229-264 ◽  
Author(s):  
Nariyuki Minami
Keyword(s):  
Asymptotic Expansion ◽  
Probability Distribution ◽  
Mild Condition ◽  
Conditional Distribution ◽  
Joint Probability ◽  
Statistical Properties ◽  
Joint Probability Distribution ◽  
Original Tree ◽  
Total Progeny

Let Y k (ω) (k ≥ 0) be the number of vertices of a Galton-Watson tree ω that have k children, so that Z(ω) := ∑ k≥0 Y k (ω) is the total progeny of ω. In this paper, we will prove various statistical properties of Z and Y k . We first show, under a mild condition, an asymptotic expansion of P(Z = n) as n → ∞, improving the theorem of Otter (1949). Next, we show that Y k (ω) := ∑ j=0 k Y j (ω) is the total progeny of a new Galton-Watson tree that is hidden in the original tree ω. We then proceed to study the joint probability distribution of Z and Y k k , and show that, as n → ∞, Y k /n k is asymptotically Gaussian under the conditional distribution P(· | Z = n).

scholarly journals On the number of vertices with a given degree in a Galton-Watson tree

2005 ◽  
Vol 37 (1) ◽  
pp. 229-264 ◽  
Author(s):  
Nariyuki Minami
Keyword(s):  
Asymptotic Expansion ◽  
Probability Distribution ◽  
Mild Condition ◽  
Conditional Distribution ◽  
Joint Probability ◽  
Statistical Properties ◽  
Joint Probability Distribution ◽  
Original Tree ◽  
Total Progeny

Let Yk(ω) (k ≥ 0) be the number of vertices of a Galton-Watson tree ω that have k children, so that Z(ω) := ∑k≥0Yk(ω) is the total progeny of ω. In this paper, we will prove various statistical properties of Z and Yk. We first show, under a mild condition, an asymptotic expansion of P(Z = n) as n → ∞, improving the theorem of Otter (1949). Next, we show that Yk(ω) := ∑j=0kYj(ω) is the total progeny of a new Galton-Watson tree that is hidden in the original tree ω. We then proceed to study the joint probability distribution of Z and Ykk, and show that, as n → ∞, Yk/nk is asymptotically Gaussian under the conditional distribution P(· | Z = n).

scholarly journals STATISTICAL PROPERTIES OF THE INTERBEAT INTERVAL CASCADE IN HUMAN HEARTS

2006 ◽  
Vol 17 (04) ◽  
pp. 571-580 ◽  
Author(s):  
FATEMEH GHASEMI ◽  
J. PEINKE ◽  
M. REZA RAHIMI TABAR ◽  
MUHAMMAD SAHIMI
Keyword(s):  
Probability Distribution ◽  
Diffusion Coefficients ◽  
Probability Distributions ◽  
Joint Probability ◽  
Planck Equation ◽  
Statistical Properties ◽  
Kolmogorov Equation ◽  
Joint Probability Distribution ◽  
Conditional Probability Distribution ◽  
And Diffusion

Statistical properties of interbeat intervals cascade in human hearts are evaluated by considering the joint probability distribution P (Δx2, τ2; Δx1, τ1) for two interbeat increments Δx1and Δx2of different time scales τ1and τ2. We present evidence that the conditional probability distribution P (Δx2, τ2| Δx1, τ1) may be described by a Chapman–Kolmogorov equation. The corresponding Kramers–Moyal (KM) coefficients are evaluated. The analysis indicates that while the first and second KM coefficients take on well-defined and significant values, the higher-order coefficients in the KM expansion are small. As a result, the joint probability distributions of the increments in the interbeat intervals are described by a Fokker–Planck equation, with the first two KM coefficients acting as the drift and diffusion coefficients. The method provides a novel technique for distinguishing two classes of subjects, namely, healthy ones and those with congestive heart failure, in terms of the drift and diffusion coefficients which behave differently for two classes of the subjects.

scholarly journals The Degree Analysis of an Inhomogeneous Growing Network with Two Types of Vertices

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Huilin Huang
Keyword(s):  
Probability Distribution ◽  
Law Of Large Numbers ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Degree Sequences ◽  
Strong Law ◽  
Large Numbers ◽  
Different Types ◽  
Azuma's Inequality ◽  
Growing Network

We consider an inhomogeneous growing network with two types of vertices. The degree sequences of two different types of vertices are investigated, respectively. We not only prove that the asymptotical degree distribution of typesfor this process is power law with exponent2+1+δqs+β1-qs/αqs, but also give the strong law of large numbers for degree sequences of two different types of vertices by using a different method instead of Azuma’s inequality. Then we determine asymptotically the joint probability distribution of degree for pairs of adjacent vertices with the same type and with different types, respectively.

Improved Locality Preserving Projection for Hyperspectral Image Classification in Probabilistic Framework

2021 ◽  
Author(s):  
Reza Seifi Majdar ◽  
Hassan Ghassemian
Keyword(s):  
Distribution Function ◽  
Probability Distribution ◽  
Probability Distribution Function ◽  
Spatial Information ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Spectral Features ◽  
Probabilistic Framework ◽  
Locality Preserving ◽  
Training Samples

Unlabeled samples and transformation matrix are two main parts of unsupervised and semi-supervised feature extraction (FE) algorithms. In this manuscript, a semi-supervised FE method, locality preserving projection in the probabilistic framework (LPPPF), to find a sufficient number of reliable and unmixed unlabeled samples from all classes and constructing an optimal projection matrix is proposed. The LPPPF has two main steps. In the first step, a number of reliable unlabeled samples are selected based on the training samples, spectral features, and spatial information in the probabilistic framework. In this way, the spectral and spatial probability distribution function is calculated for each unlabeled sample. Therefore, the spectral features and spatial information are integrated together with a joint probability distribution function. Finally, a sufficient number of unlabeled samples with the highest joint probability distribution are selected. In the second step, the selected unlabeled samples are applied to construct the transformation matrix based on the spectral and spatial information of the unlabeled samples. The adjacency graph is improved by using new weights based on spectral and spatial information. This method is evaluated on three data sets: Indian Pines, Pavia University, and Kennedy Space Center (KSC) and compared with some recent and well-known supervised, semi-supervised, and unsupervised FE methods. Various experiments demonstrate the efficiency of the LPPPF in comparison with the other FE methods. LPPPF has also considerable performance with limited training samples.

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