scholarly journals On Classes of Equivalence and Identifiability of Age-Dependent Branching Processes

2014 ◽  
Vol 46 (3) ◽  
pp. 704-718
Author(s):  
Rui Chen ◽  
Ollivier Hyrien
Keyword(s):  
Population Size ◽  
Markov Processes ◽  
Branching Processes ◽  
Biological Data ◽  
Equivalence Classes ◽  
Statistical Procedures ◽  
Age Dependent ◽  
Parametric Families ◽  
Dependent Processes

Age-dependent branching processes are increasingly used in analyses of biological data. Despite being central to most statistical procedures, the identifiability of these models has not been studied. In this paper we partition a family of age-dependent branching processes into equivalence classes over which the distribution of the population size remains identical. This result can be used to study identifiability of the offspring and lifespan distributions for parametric families of branching processes. For example, we identify classes of Markov processes that are not identifiable. We show that age-dependent processes with (nonexponential) gamma-distributed lifespans are identifiable and that Smith-Martin processes are not always identifiable.

On Classes of Equivalence and Identifiability of Age-Dependent Branching Processes

2014 ◽  
Vol 46 (03) ◽  
pp. 704-718
Author(s):  
Rui Chen ◽  
Ollivier Hyrien
Keyword(s):  
Population Size ◽  
Markov Processes ◽  
Branching Processes ◽  
Biological Data ◽  
Equivalence Classes ◽  
Statistical Procedures ◽  
Age Dependent ◽  
Parametric Families ◽  
Dependent Processes

Age-dependent branching processes are increasingly used in analyses of biological data. Despite being central to most statistical procedures, the identifiability of these models has not been studied. In this paper we partition a family of age-dependent branching processes into equivalence classes over which the distribution of the population size remains identical. This result can be used to study identifiability of the offspring and lifespan distributions for parametric families of branching processes. For example, we identify classes of Markov processes that are not identifiable. We show that age-dependent processes with (nonexponential) gamma-distributed lifespans are identifiable and that Smith-Martin processes are not always identifiable.

On a functional equation for general branching processes

1973 ◽  
Vol 10 (1) ◽  
pp. 198-205 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Population Size ◽  
General Class ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variable ◽  
Slow Variation ◽  
Age Dependent ◽  
Class Of Functions

If Z(t) denotes the population size in a Bellman-Harris age-dependent branching process such that a non-denenerate random variable W, then it is known that E(W) = 1 and that ϕ (u) = E(e–uW) satisfies a well-known integral equation. In this situation Athreya [1] has recently found a NASC for E(W |log W| y) <∞, for γ > 0. This paper generalizes Athreya's results in two directions. Firstly a more general class of branching processes is considered; secondly conditions are found for E(W 1 + βL(W)) < ∞ for 0 β < 1, where L is one of a class of functions of slow variation.

A stochastic population projection system based on general age-dependent branching processes

1987 ◽  
Vol 24 (1) ◽  
pp. 1-13
Author(s):  
Charles J. Mode ◽  
Marc E. Jacobson ◽  
Gary T. Pickens
Keyword(s):  
Population Size ◽  
Branching Processes ◽  
Population Projections ◽  
Initial Population ◽  
Projection System ◽  
Stochastic Fluctuations ◽  
Age Dependent ◽  
Initial Population Size ◽  
The Mean ◽  
Mean Function

Algorithms for a stochastic population process, based on assumptions underlying general age-dependent branching processes in discrete time with time inhomogeneous laws of evolution, are developed through the use of a new representation of basic random functions involving birth cohorts and random sums of random variables. New algorithms provide a capability for computing the mean age structure of the process as well as variances and covariances, measuring variation about means. Four exploratory population projections, testing the implications of the algorithms for the case of time-homogeneous laws of evolution, are presented. Formulas extending mean and variance functions for unit population projections to an arbitrary initial population size are also presented. These formulas show that, in population processes with non-random laws of evolution, stochastic fluctuations about the mean function are negligible when initial population size is large. Further extensions of these formulas to the case of randomized laws of evolution suggest that stochastic fluctuations about the mean function can be significant even for large initial populations.

Asymptotic growth of a class of size-and-age-dependent birth processes

1974 ◽  
Vol 11 (2) ◽  
pp. 248-254 ◽  
Author(s):  
W. A. O'N. Waugh
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Total Population ◽  
Asymptotic Expression ◽  
Branching Processes ◽  
Total Population Size ◽  
Stochastic Population Models ◽  
Age Dependent ◽  
Birth Processes ◽  
Special Case

A class of binary fission stochastic population models is described, in which the fission probabilities may depend on the age of an individual and the total population size. Age-dependent binary branching processes with Erlangian lifelength distributions are a special case. An asymptotic expression for the growth of the population size is developed, which generalizes known theorems about the asymptotic exponential growth of a branching process.

A stochastic population projection system based on general age-dependent branching processes

1987 ◽  
Vol 24 (01) ◽  
pp. 1-13
Author(s):  
Charles J. Mode ◽  
Marc E. Jacobson ◽  
Gary T. Pickens
Keyword(s):  
Population Size ◽  
Branching Processes ◽  
Population Projections ◽  
Initial Population ◽  
Projection System ◽  
Stochastic Fluctuations ◽  
Age Dependent ◽  
Initial Population Size ◽  
The Mean ◽  
Mean Function

Algorithms for a stochastic population process, based on assumptions underlying general age-dependent branching processes in discrete time with time inhomogeneous laws of evolution, are developed through the use of a new representation of basic random functions involving birth cohorts and random sums of random variables. New algorithms provide a capability for computing the mean age structure of the process as well as variances and covariances, measuring variation about means. Four exploratory population projections, testing the implications of the algorithms for the case of time-homogeneous laws of evolution, are presented. Formulas extending mean and variance functions for unit population projections to an arbitrary initial population size are also presented. These formulas show that, in population processes with non-random laws of evolution, stochastic fluctuations about the mean function are negligible when initial population size is large. Further extensions of these formulas to the case of randomized laws of evolution suggest that stochastic fluctuations about the mean function can be significant even for large initial populations.

On a functional equation for general branching processes

1973 ◽  
Vol 10 (01) ◽  
pp. 198-205 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Population Size ◽  
General Class ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variable ◽  
Slow Variation ◽  
Age Dependent ◽  
Class Of Functions

If Z(t) denotes the population size in a Bellman-Harris age-dependent branching process such that a non-denenerate random variable W, then it is known that E(W) = 1 and that ϕ (u) = E(e–uW ) satisfies a well-known integral equation. In this situation Athreya [1] has recently found a NASC for E(W |log W| y ) &lt;∞, for γ &gt; 0. This paper generalizes Athreya's results in two directions. Firstly a more general class of branching processes is considered; secondly conditions are found for E(W 1 + βL(W)) &lt; ∞ for 0 β &lt; 1, where L is one of a class of functions of slow variation.

Asymptotic growth of a class of size-and-age-dependent birth processes

1974 ◽  
Vol 11 (02) ◽  
pp. 248-254 ◽  
Author(s):  
W. A. O'N. Waugh
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Total Population ◽  
Asymptotic Expression ◽  
Branching Processes ◽  
Total Population Size ◽  
Stochastic Population Models ◽  
Age Dependent ◽  
Birth Processes ◽  
Special Case

A class of binary fission stochastic population models is described, in which the fission probabilities may depend on the age of an individual and the total population size. Age-dependent binary branching processes with Erlangian lifelength distributions are a special case. An asymptotic expression for the growth of the population size is developed, which generalizes known theorems about the asymptotic exponential growth of a branching process.

Multi-type age-dependent branching processes as models of metastasis evolution

2019 ◽  
Vol 35 (3) ◽  
pp. 284-299
Author(s):  
Maroussia Slavtchova-Bojkova ◽  
Kaloyan Vitanov
Keyword(s):  
Branching Processes ◽  
Age Dependent
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