approximate_moment(n, fun; p=ones(n)/n, q=1, m=1, r=1,
            steps=1000, normalize=true, ϵ = 1e-3)

Calculates the q-th rising moment of T[N] (number of designs that are needed to collect all modules m times). Integral is approximated by the Riemann sum.

Reference:

• Doumas, A. V., & Papanicolaou, V. G. (2016). The coupon collector’s problem revisited: generalizing the double Dixie cup problem of Newman and Shepp. ESAIM: Probability and Statistics, 20, 367-399.

Examples

julia> n = 100
julia> fun = exp_ccdf
julia> approximate_moment(n, fun; p=ones(n)/n, q=1, m=1, r=1,
steps=10000, normalize=true)
518.8175339489885

exp_ccdf(n, T; p=ones(n)/n, m=1, r=1, normalize=true)

Calculates 1 - F(t), which is the complement of the success probability F(t)=P(T ≤ t) (= probability that the expected minimum number of designs T is smaller than t in order to see each module at least m times). This function serves as the integrand for calculating E[T].

• n: number of modules in the design space
• p: vector with the probabilities/abundances of the different modules in the design space during library generation
• T: number of designs
• m: number of times each module has to observed in the sampled set of designs
• r: number of modules per design
• normalize: if true, normalize p

References:

• Doumas, A. V., & Papanicolaou, V. G. (2016). The coupon collector’s problem revisited: generalizing the double Dixie cup problem of Newman and Shepp. ESAIM: Probability and Statistics, 20, 367-399.
• Boneh, A., & Hofri, M. (1997). The coupon-collector problem revisited—a survey of engineering problems and computational methods. Stochastic Models, 13(1), 39-66.

Examples

julia> n = 100
julia> t = 500
julia> exp_ccdf(n, t; p=ones(n)/n, m=1, r=1, normalize=true)
0.4913906004535237

expectation_fraction_collected(n::Integer, t::Integer; p=ones(n)/n, r=1, normalize=true)

Calculates the fraction of all modules that is expected to be observed after collecting t designs.

• n: number of modules in design space
• t: sample size/number of designs
• p: vector with the probabilities or abundances of the different modules
• r: number of modules per design
• normalize: if true, normalize p

References:

• Boneh, A., & Hofri, M. (1997). The coupon-collector problem revisited—a survey of engineering problems and computational methods. Stochastic Models, 13(1), 39-66.

Examples

julia> n = 100
julia> t = 200
julia> expectation_fraction_collected(n, t; p=ones(n)/n, r=1, normalize=true)
0.8660203251420364

expectation_minsamplesize(n; p=ones(n)/n, m=1, r=1, normalize=true)

Calculates the expected minimum number of designs E[T] to observe each module at least m times.

• n: number of modules in the design space
• p: vector with the probabilities or abundances of the different modules
• m: number of times each module has to be observed in the sampled set of designs
• r: number of modules per design
• normalize: if true, normalize p

References:

• Doumas, A. V., & Papanicolaou, V. G. (2016). The coupon collector’s problem revisited: generalizing the double Dixie cup problem of Newman and Shepp. ESAIM: Probability and Statistics, 20, 367-399.
• Boneh, A., & Hofri, M. (1997). The coupon-collector problem revisited—a survey of engineering problems and computational methods. Stochastic Models, 13(1), 39-66.

Examples

julia> n = 100
julia> expectation_minsamplesize(n; p=ones(n)/n, m=1, r=1, normalize=true)
518

Computes the log of factorial(n), falls back on Stirling's approximation for n > 20

prob_occurrence_module(pᵢ, t::Integer, r, k::Integer)

Calculates probability that specific module with module probability pᵢ has occurred k times after collecting t designs.

Sampling processes of individual modules are assumed to be independent Poisson processes.

• pᵢ: module probability
• t: sample size/number of designs
• k: number of occurrence

References:

• Boneh, A., & Hofri, M. (1997). The coupon-collector problem revisited—a survey of engineering problems and computational methods. Stochastic Models, 13(1), 39-66.

Examples

julia> pᵢ = 0.005
julia> t = 500
julia> k = 2
julia> r = 1
julia> prob_occurrence_module(pᵢ, t, r, k)
0.25651562069968376

std_minsamplesize(n::Integer; p=ones(n)/n, m::Integer=1, r=1, normalize=true)

Calculates the standard deviation on the minimum number of designs to observe each module at least m times.

• n: number of modules in the design space
• p: vector with the probabilities or abundances of the different modules
• m: number of complete sets of modules that need to be collected
• r: number of modules per design
• normalize: if true, normalize p

Examples

julia> n = 100
julia> std_minsamplesize(n; p=ones(n)/n, m=1, r=1, normalize=true)
126

success_probability(n::Integer, t::Integer; p=ones(n)/n, m::Integer=1, r=1, normalize=true)

Calculates the success probability F(t) = P(T ≤ t) or the probability that the minimum number of designs T to see each module at least m times is smaller than t.

• n: number of modules in design space
• t: sample size/number of designs for which to calculate the success probability
• p: vector with the probabilities or abundances of the different modules
• m: number of complete sets of modules that need to be collected
• r: number of modules per design
• normalize: if true, normalize p

References:

• Boneh, A., & Hofri, M. (1997). The coupon-collector problem revisited—a survey of engineering problems and computational methods. Stochastic Models, 13(1), 39-66.

Examples

julia> n = 100
julia> t = 600
julia> success_probability(n, t; p=ones(n)/n, m=1, r=1, normalize=true)
0.7802171997092149